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Optimal Reconﬁgurable Intelligent Surface

Placement in Millimeter-Wave Communications

Konstantinos Ntontin∗, Dimitrios Selimis∗, Alexandros-Apostolos A. Boulogeorgos†, Antonis Alexandridis∗,

Aris Tsolis∗, Vasileios Vlachodimitropoulos∗, and Fotis Lazarakis∗

∗Institute of Informatics and Telecommunications, National Centre for Scientiﬁc Research "Demokritos", Greece,

e-mail: {konstantinos.ntontin, dselimis, aalex, atsolis, vvlachod, ﬂaz }@iit.demokritos.gr

†Department of Digital Systems, University of Piraeus, Greece, e-mail: al.boulogeorgos@ieee.org

Abstract—In this work, we examine the use of recon-

ﬁgurable intelligent surfaces (RISs) to create alternative

paths from a transmitter to a receiver in millimeter-wave

(mmWave) networks, when the direct link is blocked. In

this direction, we evaluate the end-to-end signal-to-noise

ratio (SNR) expression of the transmitter-RIS-receiver

links that take into account the transmitter-RIS and RIS-

receiver distances and enables us to acquire important

insights regarding the RIS position that maximizes it.

Finally, the insights are corroborated by numerical results.

I. INTRODUCTION

Increasing data-rate demands have led current mobile-access

networks relying on sub-6 GHz bands reach their limits in

terms of available bandwidth. This bottleneck created the need

to examine above-6 GHz bands for mobile-access networks.

Currently, bands in the lower-end mmWave spectrum are used

for point-to-point and point-to-multipoint line-of-sight (LOS)

wireless backhaul/fronthaul and ﬁxed-wireless access net-

works [1]. Such deployments span the 30-100 GHz operational

frequency range. However, the expected migration of future

mobile-access networks to the 30-100 GHz band pushes the

corresponding wireless backhaul/fronthaul links towards the

beyond-100 GHz bands. Due to this, backhauling/fronthauling

transceiver equipment vendors have performed LOS trials in

the D band (130-174.8 GHz), which showcase the potential of

using it in such deployments [2]. Apart from LOS, street-level

deployments in dense urban scenarios necessitate devising

non-LOS (NLOS) solutions since LOS links may not always

be ensured. However, despite the fact that according to mea-

surements [3], [4] NLOS communication through scattering

and reﬂection from objects in the radio path is feasible in the

30-100 GHz range, the higher propagation loss of beyond-100

GHz bands is likely to put such communication into question.

The conventional approach of counteracting NLOS links

is by providing alternative LOS routes through relay nodes

[5]. Although this is a well-established method to increase the

coverage when the signal quality of the direct links is low, it is

argued that it cannot constitute a viable approach for massive

deployment, especially for mmWave networks. This is due to

the increased power consumption of the active radio-frequency

(RF) components in high frequencies that relays need to

be equipped with [6]. Apart from relaying, communication

through passive non-reconﬁgurable specular reﬂectors, such as

dielectric mirrors, has been proposed as another alternative.

Such a method for coverage enhancement has the potential

to be notably more cost efﬁcient compared with relaying

and has been documented at both mmWave and beyond-100

GHz bands [6], [7]. Due to the highly dynamic nature of

blockage at high frequencies, it would be desirable that such

reﬂectors can change the angle of departure of the waves

based on changing blockage conditions so that they direct the

beams towards different routes. However, passive reﬂectors

are incapable of supporting the aforementioned functionality

since the conventional Snell’s law applies. Based on the above,

an intriguing question that arises is: Would it be possible to

deploy reconﬁgurable reﬂectors that can arbitrarily steer the

impinging beam based on dynamic blockage conditions? The

answer is afﬁrmative by considering the RIS paradigm.

RISs are two-dimensional structures of dielectric material,

in which tunable reﬂecting elements are embedded [8–11].

They constitute a substantially different technology than re-

laying, owing to their independence from bulky and power-

hungry analog electronic components, such as power ampli-

ﬁers and low-noise ampliﬁers. Additionally, their operation,

in contrast with relaying, does not require dividers and com-

biners, which can incur high insertion losses. By individually

tuning the phase response of each individual RIS element, the

reﬂected signals can constructively aggregate at a particular

focal point, such as the receiver. Such a tuning can be

enabled by electronic phase-switching components, such as

PIN diodes, RF-microelectromechanical systems, and varactor

diodes, that are introduced between adjacent elements [12].

Hence, RISs offer an alternative-to-relaying method for large-

scale beamforming without the incorporation of high power

consuming electronics and insertion losses involved by the ad-

ditional circuitry. In practice, the RIS element phase shift can

be controlled by a central controller through programmable

software [12].

Motivation and contribution: Previous works on RISs

mostly consider NLOS channels with respect to the

transmitter-RIS and RIS-receiver links, which make them

rather suitable for sub-6 GHz omnidirectional communica-

tions. Likewise, the employed models assume that the whole

RIS surface is illuminated regardless of the transmitter-RIS

distance. However, due to the fact that RISs in future networks

are expected to cover large portions of sizeable structures, such

as buildings, only a portion of the total RIS area would be illu-

minated, which depends on the aforementioned distance. Such

a reasoning necessitates the derivation of a novel expression

of the end-to-end SNR that incorporates such a dependence.

Based on the above, in this work we derive the corresponding

SNR expression in a mmWave RIS-aided link and use it to

extract important insights regarding the RIS placement that

maximizes the resulting expression. Furthermore, the ﬁndings

are validated by means of numerical simulations.

The rest of this work is structured as follows: In Section

II, we present the scenario under consideration together with

its main assumptions. In Section III, we derive the resulting

end-to-end SNR expression, which we use to extract important

insights regarding the optimal positioning of the RIS. Numeri-

cal results that validate the theoretical ﬁndings are provided in

Section IV, whereas Section V concludes this work and gives

ideas for future work.

II. SYSTEM MODEL

In this section, we ﬁrst present the scenario under consid-

eration and, subsequently, the main assumptions.

A. Scenario

We consider a communication between a street-level trans-

mitter and receiver, as depicted in Fig. 1, in a mmWave highly

directional link1. The transmitter and receiver are equipped

with large, with respect to the wavelength, antennas that enable

communication through highly directive beams. Furthermore,

we assume that due to a ﬁxed or moving obstacle the direct

communication is not possible, which can be the case for

highly directive beams and large carrier frequencies. In such

a case, the transmitter redirects the beam to an RIS mounted

on some ﬁxed structure, such a tall building. The RIS acts a

beamformer by adjusting the phase response of the elements

so that the beam is directed towards the receiver. Moreover, we

consider that both the transmitter-RIS and RIS-receiver links

are subject to LOS conditions and that the RIS is located in

the far ﬁeld of both the transmit and receive antennas.

B. Main Assumptions

We consider that the size of the RISs in the network is

sufﬁciently large so that the area illuminated by the main

lobe of a transmitted beam is smaller than RIS area. Such

an assumption is justiﬁed by the fact that it is expected that

RISs in future networks are going to cover large portions

of the building facades. In addition, their size needs to be

sufﬁciently large so that they can simultaneously accommodate

the transmissions of different transmitters. By considering

that under pencil-beam transmissions almost all the transmit

energy is located in the half-power beamwidth area (HPBW)

of the main lobe [6] and that the main-lobe shape is conical,

1As pointed out in Section I, such a link could be a ﬁxed point-to-point

link of an upcoming beyond-100 GHz backhaul/fronthaul network.

Obstacle

Transmitter Receiver

RIS

Fig. 1: Communication through an RIS.

RIS

Obstacle

rRIS

φΤx

rTx,RIS

Transmitter

Receiver

Fig. 2: Illuminated RIS area.

approximately a circular area of the RIS is illuminated by the

transmission2with radius rRIS , given by

rRIS =tan φT x

2rT x,RIS ,(1)

where φT x is the HPBW of the transmitted beam and rT x,RIS

is the distance between the centers of the transmitter and

the RIS, as depicted in Fig. 2. By assuming, without loss of

generality, a parabolic reﬂector as a transmit antenna with a

diameter DT x, it approximately holds that [13]

φT x ≈1.22 λ

DT x

.(2)

Consequently, the illuminated RIS area, which is denoted

by SRIS , can be approximated by

SRIS ≈πr2

RIS .(3)

Finally, we assume that the received signal is subject to

additive white Gaussian noise. Its power level in dBm for a

bandwidth W, denoted by N0, is equal to

N0=−174 + 10 log10 (W) + FdB,(4)

2The actual shape of the illuminated RIS area depends on the angle of

incidence of the impinging wave on the RIS. It is a circle for angle of incidence

equal to 0◦and an ellipse for angles greater than 0◦, according to the theory

of conical cross sections. However, in this work our primary focus is on the

resulting SNR expression and the optimal RIS positioning based on it, where

the latter does not depend on the shape of the RIS illuminated area. Due to

this, we consider the circular footprint model in order to simplify the ﬁnal

SNR expression.

where FdB is the noise ﬁgure in dB and Wis the transmission

bandwidth [14].

III. SNR AND OPTIMAL RIS PLACE ME NT

In this section, we ﬁrstly compute the received SNR of the

considered RIS-aided communication. Subsequently, based on

the expression, we comment on the optimal RIS placement.

Proposition 1: By assuming that all the RIS elements have

the same amplitude reﬂection coefﬁcient, denoted by Γ, the

maximum received power, which we denote by Prmax , can be

approximated as

Prmax ≈

PT xtan4φT x

2Γ2GT xGRx cos θ(arr )

0cos θ(dep)

0

16

×rT x,RIS

rRIS,Rx 2

,(5)

where PT x is the transmit power. GT x is the gain of the

transmit antenna, GRx is the gain of the receive antenna, θ(arr)

0

is the incidence angle on the RIS, θ(dep)

0is the departure angle

from the RIS, and rRIS,Rx is the distance between the centers

of the RIS and the receiver.

Proof : See Appendix A.

Assuming parabolic reﬂector antennas at the transmitter and

the receiver with diameters DTx and DRx, respectively, it

holds that

Gm=πDm

λ2

em, m ={T x, Rx},(6)

where eT x and eRx are the aperture efﬁciencies of the transmit

and receive parabolic reﬂectors, respectively.

As a result, the equivalent end-to-end SNR at the receiver

can be approximated as

SN R ≈

Ptxtan4φT x

2Γ2GT xGRx cos θ(arr )

0cos θ(dep)

0

16N0

×rT x,RIS

rRIS,Rx 2

.(7)

Remark 1: Based on (7), an important question that arrises

is whether a transmitter should focus its beam towards an

RIS that is closer to the transmitter or closer to the receiver.

From (7), we observe that the SNR expression depends on the

transmitter-RIS and RIS-receiver distances through the ratio

rT x,RIS

rRIS,Rx and the terms cos θ(arr)

0and cos θ(dep)

0. The latter

holds since the angles of incidence θ(arr)

0and departure θ(dep)

0

depend on rT x,RIS and rRIS,Rx. In particular, for an RIS

with axis parallel to the ground (without loss of generality)

the closer the RIS to the transmitter is, the smaller the angle

of arrival and the larger the angle of departure are, with

respect to the RIS normal. The opposite holds the closer

the RIS to the receiver is. Hence, targeting an RIS that is

closer to the transmitter or to the receiver would roughly have

the same effect on the value of the term cos θ(arr)

0cos θ(dep)

0

that is included in (7). As a result, the equivalent end-to-end

SNR expression is expected to be mainly determined by the

value of the ratio rT x,RIS

rRIS,Rx . The particular ratio reveals that the

hTx hRx

hRIS

rTX,Rx,hor

φTX

rTx,Rx

rTx,RIS rRIS,Rx

rTX,RIS,hor

θ0(dep)

θ0(arr)

Transmitter Receiver

Fig. 3: 2D simulation setup.

transmitter should target an RIS that is closer to the receiver

than the transmitter so that the equivalent end-to-end SNR is

maximized.

IV. NUMERICAL RES ULTS

The aim of this section is to validate Remark 1 regarding

the optimal RIS positioning that maximizes the end-to-end

equivalent SNR by means of simulations. Towards this, we

consider the 2D point-to-point simulation setup that is depicted

in Fig. 3, in which the axis of the RIS is parallel to the ground.

According to the considered geometry, it holds that

rT x,RIS =qr2

T x,RIS hor + (hRIS −hT x )2,

rRIS,Rx =q(rT x,Rx,hor −rT x,RIS hor )2+ (hRIS −hRx )2,

θ(arr)

0=tan−1rT x,RIS,hor

hRIS −hT x ,

θ(dep)

0=tan−1|rT x,Rx,hor −rT x,RI Shor |

hRIS −hRx .

(8)

TABLE I: Parameter values used in the simulation.

f140 GHz

PT x 1 W

W2 GHz

FdB 10 dB

rT x,Rx,hor 40 m

hRIS 15 m

hT x,hRx 3 m

DT x,DRx 10 cm

eT x,eRx 1

Γ0.9

In Table I, we present the considered values for the involved

parameters in the simulation3.

3The value hRIS = 15 m can be a typical value for an RIS that is located

in the top of a 5-ﬂoor building facade by considering that the height of each

ﬂoor is around 3 m. In addition, the values hT x =hRx = 3 m can typically

correspond to transmitter and receiver nodes that are mounted on lampposts.

0 10 20 30 40 50 60

90

95

100

105

110

115

120

rTx,RIS,hor [m]

SNR [dB]

Fig. 4: SNR vs. the horizontal transmitter-RIS distance.

0 10 20 30 40 50 60

0

0.5

1

1.5

2

2.5

rTx,RIS,hor [m]

SRIS [m2]

Fig. 5: RIS area vs. the horizontal transmitter-RIS distance.

Regarding the optimal direction of towards a large RIS

surface that the transmitter should focus its beam on or,

from another perspective, the optimal placement of an RIS,

in Fig. 4 we illustrate the SNR as a function of the horizontal

transmitter-RIS distance, where the SNR is computed accord-

ing to (7). As we observe from Fig. 4, for SNR maximization

the transmitter should focus its beam on a point on the RIS

area that is very close to the receiver. Hence, Remark 1 is

validated. Moreover, we observe that substantial SNR gains,

in particular more than 20 dB, are achieved by the optimal

beam focusing compared with, for instance, the focusing on

the point on the RIS area that is vertically above the transmitter

(rT x,RIS,hor = 0). In addition, in Fig. 5 we illustrate SRIS as a

function of the horizontal transmitter-RIS distance. We observe

that at the distance for which the maximum SNR is obtained

the corresponding illuminated RIS area is approximately equal

to 1 m2. Consequently, this showcases the feasibility of such

a deployment since RIS surfaces with areas much larger than

1m2can be installed onto large building facades. Finally, we

note that in the simulated scenario the illuminated RIS area

is located in the far ﬁeld of both the transmit and receive

antennas, as it is required in our model. This is due to the

fact that the Fraunhofer distance is equal to 2D2

m/λ = 10 m,

m={T x, Rx}, for both the transmit and receive antennas

and the minimum distance between the possible RIS location

and the aforementioned antennas is equal to 12 m.

V. CONCLUSIONS

We have conducted this work in order to determine the

optimal RIS placement with respect to the transmitter and

receiver antenna positioning. Towards this, we have considered

a realistic mmWave scenario in which the RIS area that is

illuminated by the transmitted very directional beam is smaller

than the total RIS area, which is expected in forthcoming RIS

deployments. Based on this model, we have quantiﬁed the

equivalent end-to-end SNR, which reveals that the RIS should

be optimally placed closer to the receiver than the transmitter

so that the SNR is maximized. Such an analytical outcome

was validated by means of simulations, which showed that

substantial gains are expected by the optimal deployment.

Future work will focus on the comparison of an RIS- and a

relay-aided point-to-point scenario in terms of coverage prob-

ability and rate, where the optimal deployments are considered

for both cases.

ACKNOWLEDGEMENTS

This work has reveived funding from the H2020 ARIADNE

project (Grant Agreement no. 871464).

APPENDIX

A. Proof of Proposition 1

The incident wave on the RIS nth element is given by

Einc,n =Eince−j2π

λrT x,RISnˆ

ax,(9)

where rT x,RISnis the distance between the transmitter and

the nth element of the illuminated RIS area. Considering that

the spatial power density of the incident electric wave on the

nth element of the RIS is approximately4equal to PT xGT x

4πr2

T x,RIS

,

it holds that E2

inc

2η=PT xGT x

4πr2

T x,RIS

,(10)

where ηis the free-space impedance. The captured power by

the nth element, denoted by Pcap,n, of the RIS is given by

Pcap,n =E2

inc

2ηAeff ,n =λ

4π2PT xGT x GRIS,n θ(arr)

0

r2

T x,RIS

,(11)

where Aeff,n =λ2

4πGRIS,n θ(arr)

0is the effective aperture

of the nth RIS element with GRIS,n θ(arr)

0being its gain.

Based on the passive reﬂector theory, it holds that [13]

GRIS,n θ(arr )

0=4π

λ2dxdycos θ(arr)

0,(12)

where dxand dyare the x and y-axis dimensions of each RIS

element, respectively. The term dxdycos θ(arr)

0represents

the projected surface on the RIS seen by the transmitter.

4Considering that rT x,RISn≈rT x,RI S due to the far-ﬁeld operation.

The spatial power density at the receiver antenna after re-

ﬂection from the nth element, which is denoted by Prspatial,n ,

is given by

Prspatial,n =Pcap,n

Γ2GRIS,n θ(dep)

01

4πr2

RIS,Rx

=λ2

(4π)3

PT xΓ2GT x GRIS,n θ(arr)

0GRIS,n θ(dep)

0

r2

T x,RIS r2

RIS,Rx

,

(13)

where, as in the transmission towards the RIS case, it is

assumed that due to the far-ﬁeld positioning of the receiver

antenna with respect to the RIS it holds that rRISn,Rx ≈

rRIS,Rx , where rRISn,Rx is the distance from the center of

the nth RIS element to the center of the receiver antenna.

In addition, GRIS,n θ(dep)

0is the gain of the nth RIS

element involved in the radiation towards the receiver antenna.

Similarly to (12), it holds that

GRIS,n θ(dep)

0=4π

λ2dxdycos θ(dep)

0,(14)

where the term dxdycos θ(dep)

0represents the projected

surface on the RIS seen by the receiver.

Next, by assuming that the nth RIS element through its

corresponding switching element is adjusted so that the phase

of the departing beam related with the element is equal to

φRIS nand that mutual coupling among different elements

is negligible, the electric ﬁeld of the impinging beam at the

receiver antenna, which we denote by Er,n, is given by

Er,n =Er,ne

−j φRISn+2π(rT x,RI Sn+rRISn,Rx )

λ!ˆ

ax,(15)

where

Er,n =q2ηPrspatial,n

=v

u

u

t2ηλ2

(4π)3

PT xΓ2GT x GRIS,n θ(arr)

0GRIS,n θ(dep)

0

r2

T x,RI S r2

RIS,Rx

.

(16)

The received power PrφRIS1, ..., φRISMxMyconsidering

the contribution from all the RIS elements is given by

P(RIS )

rφRIS1, ..., φRI SMxMy

=

PMxMy

n=1 Er,n

2

2η

λ2

4πGRx

=λ

4π4PT xΓ2GT x GRx

r2

T x,RIS r2

RIS,Rx

×

MxMy

X

n=1 rGRIS,n θ(arr)

0GRIS,n θ(dep)

0

×e

−j φRISn+2π(rT x,RI Sn+rRISn,Rx )

λ!

2

,(17)

where Mxand Myare the number of RIS elements in the x-

and y-axis, respectively.

From (17), it is evident that the received power is maximized

by setting

φRISn=−2π(rT x,RI Sn+rRISn,Rx )

λ.(18)

By taking into account that

MxMy

X

n=1 rGRIS,n θ(arr)

0GRIS,n θ(dep)

0

2

=GRIS θ(arr )

0GRIS θ(dep)

0,(19)

where

GRIS θ(s)

0=4π

λ2SRIS cos θ(s)

0, s ={arr, dep},(20)

and by using (1) and (3), the maximum value of

PrφRIS1, ..., φRI SMxMy,Prmax , is approximated by (5),

which concludes the proof.

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